We study a variant of the uncertainty principle in terms of the annihilation and creation operator on generalized Segal Bargmann spaces, which are used for the FBI-Bargmann transform. In addition, we compute the Berezin transform of these operators and indicate how to use spaces of entire functions in one variable to study the Szego kernel for hypersurfaces in C-2.

In this paper we analyse the domains of differential and multiplication operators on the Segal-Bargmann space. We consider the basic estimate for the partial derivative-complex, remark that this estimate is closely related to the uncertainty principle in quantum mechanics and compute the Bergman kernel of the graph norm. It is shown that the set of all functions u in the Segal-Bargmann Lambda(2)(C, e(-|z|2)) such that the multiplication with a polynomial p is norm bounded gives a relatively compact subset of the Segal-Bargmann space. In the following section we give a survey of recent results on the partial derivative-complex on weighted Bergman spaces on Hermitian mani- folds, analysing metrics which produce a similar duality between differentiation and multiplication as in the Segal-Bargmann space. Finally we study the basic estimates and the corresponding questions of compactness for the generalized partial derivative-complex.

We study certain densely defined unbounded operators on the Segal-Bargmann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding D-complex and study properties of the complex Laplacian (square) over tilde (D) = DD* + D*D, where D is a differential operator of polynomial type, in particular we discuss the corresponding basic estimates, where we express a commutator term as a sum of squared norms.

Let (M, h) be a Hermitian manifold and psi a smooth weight function on M. The partial derivative-complex on weighted Bergman spaces A((p,0))(2)(M, h, e(-psi)) of holomorphic (p, 0)-forms was recently studied in [10] and [9]. It was shown that if h is Kahler and a suitable density condition holds, the partial derivative-complex exhibits an interesting holo-morphicity/duality property when (partial derivative psi)# is holomorphic (i.e., when the real gradient field grad(h)psi is a real holomorphic vector field.) For general Hermitian metrics, this property does not hold without the holomorphicity of the torsion tensor T-p(rs). In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kahler and conformally Kahler manifolds and their relationship to the partial derivative-complex on weighted Bergman spaces. For Kahler metrics with multi-radial potential functions on C-n, we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally Kahler metrics on complex space forms, we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold S2n-1 x S-1 and the "half " hyperbolic metric on the unit ball. For some of these metrics, we further de-termine weight functions psi with real holomorphic gradient fields. They provide a wealth of triples (M, h, e(-psi)) of Hermitian non-Kahler manifolds with weights for which the partial derivative-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the partial derivative-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the partial derivative-equation.

We study certain densely defined unbounded operators on the Segal-Barg\-mann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding $D$-complex and study properties of the corresponding complex Laplacian $\tilde \Box_D = D D^* + D^* D,$ where $D$ is a differential operator of polynomial type.

Firstly we establish a sharp pointwise estimate for the arbitrary derivative of the function $f\in F_{\alpha}^{p},$ where $F_{\alpha}^{p}$ denotes the Fock space for $1\leq p \alpha$ and its adjoint.

In this paper, we investigate the partial derivative-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in C-n, studied earlier by the first-named author, in which the adjoint of the differentiation is the multiplication by z. The results are applied to two important examples in the unit ball, namely, the complex hyperbolic metric and a conformally Kahler metric which are related to Bergman spaces with so-called "exponential" and "standard" weights, respectively. In particular, we obtain new estimates for the solutions of the partial derivative-equation on these weighted Bergman spaces. (C) 2020 Elsevier Inc. All rights reserved.

We study necessary conditions for compactness of the weighted (partial derivative) over bar -Neumann operator on the space L-2 (C-n, e(-phi)) for a plurisubharmonic function phi. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non)compactness of the (partial derivative) over bar -Neumann operator for decoupled weights, which are of the form phi(z) = phi(1)(z(1)) + ... + phi(n) (z(n)). More can be said if every Delta(phi j) defines a nontrivial doubling measure.

We study certain densely defined unbounded operators on the Segal-Bargmann space. These are the annihilation and creation operators of quantum mechanics. In several complex variables we have the partial derivative-operator and its adjoint partial derivative* acting on (p, 0)-forms with coefficients in the Segal-Bargmann space. We consider the corresponding partial derivative-complex and study the spectral properties of the corresponding complex Laplacian (square) over tilde-partial derivative*partial derivative. Finally, we study a more general complex Laplacian (square) over tilde (D) - DD* + D*D, where D is a differential operator of polynomial type, to find the canonical solutions to the inhomogeneous equations Du = alpha and D*v = beta.

We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-\varphi})$, under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension. Moreover, we compute the essential spectrum of the complex Laplacian for decoupled weights, $\varphi(z) = \varphi_1(z_1) + \dotsb + \varphi_n(z_n)$, and investigate (non-) compactness of the $\overline\partial$-Neumann operator in this case. More can be said if every $\Delta\varphi_j$ defines a nontrivial doubling measure.

In this paper we characterize compactness of the canonical solution operator to ∂ on weigthed L spaces on C. For this purpose we consider certain Schrödinger operators with magnetic fields and use a condition which is equivalent to the property that these operators have compact resolvents. We also point out what are the obstructions in the case of several complex variables.

We study certain densely defined unbounded operators on the Fock space. These are the annihilation and creation operators of quantum mechanics. In several complex variables we have the ∂-operator and its adjoint ∂^* acting on (p,0)-forms with coefficients in the Fock space. We consider the corresponding ∂-complex and study spectral properties of the corresponding complex Laplacian = ∂∂^* + ∂^*∂. Finally we study a more general complex Laplacian _D = D D^* + D^* D, where D is a differential operator of polynomial type, to find the canonical solutions to the inhomogeneous equations Du=α and D^*v=β.

We apply methods from complex analysis, in particular the d-bar-Neumann operator, to investigate spectral properties of Pauli operators.

We apply methods from complex analysis, in particular the partial derivative-Neumann operator, to study spectral properties of Pauli operators. For this purpose we consider the weighted partial derivative-complex on C-n with a plurisubharmonic weight function. The Pauli operators appear at the beginning and at the end of the weighted partial derivative-complex. We use the spectral properties of the corresponding partial derivative-Neumann operator to answer the question when the Pauli operators are with compact resolvent. It is also of importance to know whether the related Bergman space of entire functions is of infinite dimension. The main results are formulated in terms of the properties of the Levi matrix of the weight function. If the weight function is decoupled, one gets additional informations. Finally, we point out that a corresponding Dirac operator fails to be with compact resolvent.

We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on b C^n. In addition we describe an approach to obtain the compactness estimates for the d-bar-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich - Kondrachov lemma.